I have thought about developing an alternate script for Norwegian/Scandinavian based on the old runes, really just because I think that would be fun and interesting, but insofar as I haven’t done this yet, I don’t know of any alternate scripts for Norwegian, and obviously there isn’t much point in talking about English.
So I will instead share an alternate script for a language I do not speak: Circassian, or specifically West Circassian or Adyghe.
Father and son duo R. I. Daur and I. Yu. Daur together developed an alternate script for that language in I think 2012. The script was dubbed “Mifo-Circassian”, and as I understand it, it’s an attempt to give the Circassian language a more unique visual identity, by using letters based on old inscribed symbols — I think the emblems of clans more specifically — rather than using an adapted foreign alphabet like Latin, Arabic, or Cyrillic. Furthermore, Circassian has a very unique inventory of sounds that interact with each other in unique ways, that foreign alphabets can’t really do justice, so this alphabet is more uniquely suited for the challenges that come with representing Circassian in writing. Mifo-Circassian writing seems like it may be used both alphabetically and alphasyllabically, but the alphabetic form is far more common.
Well, “more common”, not that many Circassians actually do use this Mifo-Circassian script to begin with. Cyrillic, Latin, and Arabic have the benefit of being supported by Unicode and of also being used by neighboring languages, and they work well enough for representing Circassian. So it seems like the primary usage of Mifo-Circassian is not necessarily for communicative writing, but rather for more ornamental or artistic usage.